Which of the following is equivalent to `(38+ 18i)/(4+6i)` ?

Options

A. `11/13 -3i`
B. `5-3i`
C. `19/2+3i`
D. `44+300i`

Answer

To solve the problem \((38 + 18i)/(4 + 6i)\), we will rationalize the denominator and simplify the expression step by step.

### Step 1: Rationalize the Denominator
We start with the expression:
\[ \frac{38 + 18i}{4 + 6i} \]
To rationalize the denominator, we multiply both the numerator and the denominator by the conjugate of the denominator, which is \(4 - 6i\):
\[ \frac{(38 + 18i)(4 - 6i)}{(4 + 6i)(4 - 6i)} \]

### Step 2: Multiply the Denominator
Now, we calculate the denominator:
\[ (4 + 6i)(4 - 6i) = 4^2 - (6i)^2 = 16 - 36(-1) = 16 + 36 = 52 \]

### Step 3: Multiply the Numerator
Next, we calculate the numerator:
\[ (38 + 18i)(4 - 6i) = 38 \cdot 4 + 38 \cdot (-6i) + 18i \cdot 4 + 18i \cdot (-6i) \]
Calculating each term:
- \(38 \cdot 4 = 152\)
- \(38 \cdot (-6i) = -228i\)
- \(18i \cdot 4 = 72i\)
- \(18i \cdot (-6i) = -108(-1) = 108\)

Now, combining these results:
\[ 152 - 228i + 72i + 108 = 152 + 108 - 228i + 72i = 260 - 156i \]

### Step 4: Combine the Results
Now we can write the entire expression:
\[ \frac{260 - 156i}{52} \]

### Step 5: Simplify the Expression
We can simplify this by dividing both terms in the numerator by the denominator:
\[ \frac{260}{52} - \frac{156i}{52} = 5 - 3i \]

### Final Answer
Thus, the expression \((38 + 18i)/(4 + 6i)\) simplifies to:
\[ \boxed{5 - 3i} \]